Pilot-Aided Coherent Receiver for Optical Communications

ABSTRACT

A method decodes an optical signal transmitted over an optical channel from a transmitter to a receiver. The receiver receives the transmitted optical signal to produce a digital signal including data symbols and pilot symbols, and determines filtering coefficients based on an error between amplitudes of the received pilot symbols and amplitudes of transmitted pilot symbols, while ignoring errors between phases of the received pilot symbols and phases of the transmitted pilot symbols. The amplitudes and the phases of the transmitted pilot symbols are known at the transmitter and the receiver. The receiver filters the digital signal according to the filtering coefficients to produce a filtered signal with equalized amplitude and an unconstrained phase demodulates and decodes the filtered signal to produce an estimate of the transmitted optical signal.

RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119(e) from U.S.provisional application Ser. No. 62/134,117 filed on Mar. 17, 2015,which is incorporated herein by reference.

FIELD OF THE INVENTION

This invention relates generally to coherent optical communicationssystems, and more particularly to decoding data transmitted over anoptical channel.

BACKGROUND OF THE INVENTION

Inaccuracies in carrier-phase estimation and amplitude equalizationcause distortions, i.e., the noise enhancements, which reduce theperformance of optical communications systems. In the opticalcommunications, different algorithms are used to reduce the distortion.Those algorithms are based on a hard decision for determining the phaseand amplitude of the received signal. For example, a decision-directedleast-mean-square (LMS) method uses the hard decision for determiningthe error for the updating.

However, the hard decisions can be incorrect causing suboptimal phaseand amplitude equalization. The problem of inaccuracy of the harddecisions is especially apparent in the applications with lowsignal-to-noise ratios (SNR). However, for each fixed SNR, there is aneed to further improve the data throughput and other performancemetrics of optical communications, such as spectral efficiency of thetransmitted signal.

In order to provide higher optical interface rates, recent research hasfocused on the expansion of both bandwidth and spectral efficiency.While some researches have focused on the slicing of the receivedsignals in the time or frequency domains, these solutions requireseveral parallel coherent receivers. Current results using a singlecoherent receiver have exceeded 640 Gb/s net bit rate. However, there isa demand to provide a system and a method for detection of a net bitrate in excess of 1 Tb/s with a single coherent receiver.

Detection of the bit rates in excess of 1 Tb/s with a single receiverrequires accurate demodulation of the signals. To demodulate signals inan optical communications system, it is necessary to equalizedistortions to both phase and amplitude of the received signals, causedby the optical and electrical components. This is particularly difficultfor densely modulated signals with high-order quadrature-amplitudemodulation (QAM), such as 64-QAM and 256-QAM.

Some conventional systems, such as a system described in U.S. Pat. No.8,320,778, perform amplitude equalization using a radius-directedconstant-modulus algorithm (CMA) equalizer, with an updating algorithm,such as an LMS method. That method yields acceptable results with highSNR and reasonably low density modulation, e.g., 8QAM, 16QAM. However,densely modulated signals that can result in bit rates in excess of 1Tb/s can cause significant tap noise at the equalizer output, due to theinaccuracy of the blind decisions based on the radius of the receivedsignals.

For equalization of phase distortions, conventional systems can use ablind phase search approach, see, e.g., U.S. 2011/0217043. However, thatapproach has a high complexity for densely modulated signals and suffersfrom poor performance in a low SNR regime.

SUMMARY OF THE INVENTION

Some embodiments of the invention are based on realization thatpilot-aided equalization can be advantageously used for optical signals,especially for the equalization of the densely modulated opticalsignals, such as dual-polarization (DP) 64QAM and DP-256QAM. This isbecause the decrease of the bit rate due to the pilot symbols can becompensated by the increase of the modulation order of the opticalsignals due to more accurate equalization of the optical signals.

Some embodiments of an invention are based on recognition that inoptical communications the distortion of the amplitude of the signals isslower than the distortion of the phase. Therefore, it is possible toseparate equalization of the amplitude and equalization of the phase ofthe signals transmitted over an optical channel. To that end, someembodiments perform pilot-aided amplitude equalization, while ignoringthe phase component of the pilot signals. Additionally or alternatively,some embodiments perform pilot-aided phase equalization separately fromthe amplitude equalization.

Separate amplitude and phase pilot-aided equalization used by someembodiments of the invention is in contrast with the pilot-aidedequalization attempting to correct both amplitude and phase at the sametime. While joint equalization is sufficient for wireless systems, thehigh levels of phase noise, and the need for parallel processing inoptical systems, necessitates the separation of amplitude equalizationand the estimation of carrier phase.

In addition, the separate equalization allows using multiple pilotsymbols, e.g., by averaging the error over several pilots, thereforediminishing the influence of noise on the received pilot amplitude.Accordingly, some embodiments use pilot symbols to equalize amplitude ofthe optical signal separately from the phase equalization, e.g., performamplitude equalization based on a difference of radiuses of transmittedand received pilot symbols.

Some embodiments of the invention are based on another realization thatthe phases of the optical signals are rapidly changing and are subjectto phase noise and additive noise, so just comparisons of the phases ofpilot symbols do not provide accurate results. Due to the physicalnature of the optical channel, the phases of the received pilot symbolsdepend on phases of other received symbols including other pilotsymbols. Therefore, by collectively considering the multitude of phasesof the received pilot symbols, that dependency can be used to estimatethe phase of the data symbols. Accordingly, some embodiments use pilotsymbols to determine the probability distribution of phases for theentire signals. Soft decisions and expectation maximization (EM) canthen be used to refine that probability distribution.

Accordingly, one embodiment discloses a method for decoding an opticalsignal transmitted over an optical channel from a transmitter to areceiver. The method includes receiving the transmitted optical signalto produce a digital signal including data symbols and pilot symbols;determining filtering coefficients based on an error between amplitudesof the received pilot symbols and amplitudes of transmitted pilotsymbols, while ignoring errors between phases of the received pilotsymbols and phases of the transmitted pilot symbols, wherein theamplitudes and the phases of the transmitted pilot symbols are known atthe transmitter and the receiver; filtering the digital signal accordingto the filtering coefficients to produce a filtered signal with anequalized amplitude and an unconstrained phase; and demodulating anddecoding the filtered signal to produce an estimate of the transmittedoptical signal. At least some steps of the method are performed using aprocessor of the receiver.

Another embodiment discloses a receiver for decoding an optical signaltransmitted by a transmitter over an optical channel including a frontend for receiving the transmitted optical signal to produce a digitalsignal including data symbols and pilot symbols; an amplitude equalizerfor determining filtering coefficients based on an error betweenamplitudes of the received pilot symbols and amplitudes of transmittedpilot symbols, while ignoring errors between phases of the receivedpilot symbols and phases of the transmitted pilot symbols and forfiltering the digital signal according to the filtering coefficients toproduce a filtered signal with an equalized amplitude and anunconstrained phase; a phase equalizer for determining a probabilitydistribution of phase noise on the data symbols using a statisticalprobability distribution of phase noise on the optical channel and aprobability distribution of phase noise on the pilot symbols; and adecoder for demodulating and decoding the filtered signal using theprobability distribution of phase noise on the data symbols to producean estimate of the transmitted optical signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a block diagram of a pilot-aided optical communicationssystem according to some embodiments of the invention;

FIG. 1B is an exemplar structure of a signal including a set of datasymbols and a set of pilot symbols with known amplitudes and phasesaccording to some embodiments of the invention;

FIG. 1C is a block diagram of a transmitter for a pilot-aided opticalcommunications system according to one embodiment of the invention;

FIG. 1D is a block diagram of a receiver for the pilot-aided opticalcommunications system according to one embodiment of the invention;

FIG. 2 is a flow diagram of a method for decoding an optical signalaccording to some embodiments of the invention;

FIG. 3 is a schematic of an amplitude equalizer according to someembodiments of the invention;

FIG. 4 is a schematic of the receiver for a pilot-aided opticalcommunications system according to one embodiment of the invention;

FIG. 5A is a flow diagram of a method for pilot-aided phase equalizationof the transmitted optical signal according to some embodiments of theinvention;

FIG. 5B is a block diagram of an exemplar implementation of pilot-aidedphase equalization according to one embodiment of the invention;

FIG. 5C is a block diagram of a method for determining the probabilitydistribution of phase noise on the pilot symbols according to oneembodiment of the invention;

FIG. 5D is a block diagram of a method for refining the probabilitydistribution of phase noise on the data symbols according to oneembodiment of the invention;

FIG. 5E is a flow diagram of a method for filtering the refinedprobability distribution of phase noise on the data symbols according toone embodiment of the invention;

FIG. 6 is a schematic of the structure of the pilot symbols within thedata symbols for the purpose of multi-channel phase estimation;

FIG. 7A is a block diagram of a transmitter for a multi-channelpilot-aided optical communications system according to some embodimentsof the invention;

FIG. 7B is a block diagram of a receiver for a multi-channel pilot-aidedoptical communications system according to some embodiments of theinvention; and

FIG. 8 is a schematic of the receiver for a multi-channel pilot-aidedoptical communications system incorporating training-aidedinitialization according to some embodiments of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1A shows a block diagram of a pilot-aided optical communicationssystem according to some embodiments of the invention. Data from asource (001) are sent to a transmitter (Tx) (010). For example, the dataare sent to an optional forward error correction (FEC) encoder (011) andthen the data are sent to a pilot insertion block (012), where pilotsymbols from a pilot sequence (013) are added at some pre-determinedrate to produce a signal including a set of data symbols and a set ofpilot symbols with known amplitudes and phases. After insertion of thepilot symbols, the signal undergoes digital signal processing (DSP)(014). In some embodiments, the DSP also performs other functions suchas mapping, filtering and pre-equalization. The signal is then sent tothe transmitter front end (015), where analog operations such asamplification, filtering, modulation and up-conversion occur, and thentransmitted over an optical channel (020) to a receiver (Rx) (030).

At the receiver, the signal passes through the receiver front end (031)for performing analog operations such as down-conversion, amplification,filtering and quantization of the received signal to produce a digitalsignal. The digital signal is processed by digital algorithms (032),before extraction of the received pilot symbols (033). The extractedpilot symbols are then processed in combination with the transmittedpilot sequence (035) with known amplitudes and phases corresponding tothe pilot symbols (013), by the pilot-aided DSP algorithms (036).Information resulting from this processing is then used in the receiverDSP (032) in order to improve accuracy of the equalization and carrierphase recovery. The received signal after pilot extraction is thenoptionally sent for FEC decoding (034), before being sent to adestination, e.g., a data sink (040).

FIG. 1B shows an exemplar structure of the corresponding digital signalto be encoded. The signal includes a set of data symbols and a set ofpilot symbols with known amplitudes and phases according to someembodiments of the invention. Several pilot symbols (410, 412, 414, and416) are distributed through several blocks of data symbols (411, 413,and 415). Digital signal processing on a single block of data symbols(413) can be performed using adjacent pilot symbols (412, 414) and/ornon-adjacent pilot symbols (410, 416).

FIG. 1C shows a block diagram of a transmitter for a pilot-aided opticalcommunications system according to one embodiment of the invention. Data(110) from a source (101) are sent to a transmitter (120). In thetransmitter, the data are encoded by an FEC encoder (121), before pilotsymbols (124) are inserted intermittently. The signal then undergoesprocessing with DSP algorithms and other front end electronics (122)such as analog-to-digital convertors. The signal is then sent to thetransmitter optics (123) for modulation on to the optical carrier. Theoptical signal is then sent to a wavelength multiplexer (WM) (130),where the signal can be optionally combined with other optical signalswhich have different wavelengths (131), before being sent to the opticalchannel (135).

FIG. 1D shows a block diagram of a receiver in the pilot-aided opticalcommunications system according to one embodiment of the invention. Thesignal from the optical channel (141) is sent to a wavelengthde-multiplexer (WDM) (150). Other wavelengths (151) are optionally sentto other receivers for processing independently of the wavelengthchannel of interest. The signal is then sent to the receiver (160). Theoptical signal is detected by the optical receiver front end (161). Thisblock can include both optical and electronic elements, such asdown-conversion, amplification, and quantization. The digital signal isthen processed by DSP algorithms (162). After DSP processing, thereceived pilot symbols are extracted (165), and processed (164) incombination with the known transmitted sequence (166) of pilot symbols.Information from this pilot processing is then used in the Rx DSP (162)that includes an equalizer for functions such as amplitude and phaseequalization. The processed signal is then sent for demodulation module(163) to produce soft-decision information for FEC decoding, beforebeing sent to its final destination, e.g., the data sink (170).

Amplitude Equalization

Some embodiments of the invention are based on a general realizationthat for optical communications the rate of phase variations of theoptical signal is different than the rate of variations of the amplitudeof the optical signal. Therefore, it is advantageous to equalize thephase and the amplitudes of the optical signal separately.

FIG. 2 shows a flow diagram of a method for decoding an optical signaltransmitted over an optical channel from a transmitter to a receiveraccording to some embodiments of the invention. In these embodiments,the amplitude of the transmitted optical signal is equalized separatelyfrom the phase of the transmitted optical signal. At least some steps ofthe method are performed using a processor and other components of thereceiver.

The method filters 220 the received optical signal 210, with filtercoefficients 230, to produce a digital signal including data symbols 280and pilot symbols 250. This process produces a filtered signal 235 withan equalized amplitude and an unconstrained phase, and demodulates anddecodes 240 the filtered signal to produce an estimate 245 of thetransmitted optical signal. The method then determines 260 a new set offiltering coefficients 230 based on an error between amplitudes of thereceived pilot symbols 250 and amplitudes of transmitted pilot symbols270, while ignoring errors between phases of the received pilot symbolsand phases of the transmitted pilot symbols.

FIG. 3 shows a schematic of a dual polarization amplitude equalizersuitable for a high phase noise application such as opticalcommunications, where polarization multiplexing is used according tosome embodiments of the invention. The equalizer updates the filteringcoefficients iteratively, e.g., receiving each pilot symbol and/or inresponse to determining an error in filtering the signal.

In different embodiments of the invention, the amplitudes of the pilotand/or data symbols include the amplitudes on both of the two orthogonalpolarizations used for data transmission, and the amplitudes are definedby the radiuses on each polarization. Both the amplitudes and the phasesof the transmitted pilot symbols are known at the transmitter and thereceiver.

In the amplitude equalizer, an input signal (300) is sent to a finiteimpulse response (FIR) filter (301) for equalization of amplitudedistortion, forming an output (308), i.e., a filtered signal withequalized amplitude and unconstrained phase. The coefficients of thefilter are updated as follows. Received pilot symbols are extracted(309), e.g., from the equalized signal, and their radius calculated(302). Concurrently, the known pilot symbols (306) also go through aradius calculation procedure (307). In some embodiments, thiscalculation can be performed in advance, and stored in memory.

An equalizer error is then calculated (303) based on the radius of thereceived pilot symbol and the radius of the transmitted pilot symbol,for example, using the Godard algorithm. According to some embodiments,the error term then optionally undergoes filtering in order to improveits accuracy. The error term is then used as an input to an updatingalgorithm such as the LMS algorithm or recursive least-squares (RLS)algorithm to calculate new filter coefficients (305).

For example, the outputs 308 of the 2-by-2 MIMO filter 301 are given by:

ν_(x) =h _(xx) u _(x) ^(H) +h _(yx) u _(y) ^(H), and

ν_(y) =h _(xy) u _(x) ^(H) +h _(yy) u _(y) ^(H),

where u_(x) and u_(y) are the input vectors 300 on the x and ypolarizations respectively, h_(xx), h_(yx), h_(xy) and h_(yy) are thecoefficients of the four FIR filters, ν_(x) and ν_(y) are theinstantaneous outputs 308 on the x and y polarizations respectively, andthe superscript of ^(H) operator is the Hermite transpose.

Some embodiments calculate error terms 303 according to the radiuses307, 302 of the transmitted 306 and received 309 pilots on eachpolarization, for example, according to the CMA:

e _(x) =|p _(x)|²−|ν_(x)|²

e _(y) =|p _(y)|²−|ν_(y)|²,

where e_(x) and e_(y) are the error terms on the x and y polarizationsrespectively, and p_(x) and p_(y) are the pilot symbols on the x and ypolarizations respectively.

The error can be further refined by using a filtered version of theerror term, given in the case of a sliding window accumulator filter as:

$e_{x^{\prime}} = {\sum\limits_{n = 1}^{M}\; {{e_{x}(n)}/M}}$$e_{y^{\prime}\;} = {\sum\limits_{n = 1}^{M}{{e_{y}(n)}/M}}$

where e_(x), and e_(y), are the averaged error terms on the x and ypolarizations respectively, and M is the number of error terms, whichare averaged.

The coefficients of the filter 301 are determined using the error termand some adaptation algorithm, for example, the LMS algorithm, which isdetermined by the following set of equations:

h _(xx) ′=h _(xx) +μe _(x) u _(x)ν_(x)*

h _(yx) ′=h _(yx) +μe _(x) u _(y)ν_(x)*

h _(xy) ′=h _(xy) +μe _(y) u _(x)ν_(y)*

h _(yy) ′=h _(yy) +μe _(y) u _(y)ν_(y)*,

where the vectors h_(xx)′, h_(yx)′, h_(xy)′ and h_(yy)′ are the updatedfilter coefficient vectors, the superscript of * is the conjugateoperator, μ is the equalizer convergence parameter.

The filter coefficients are updated iteratively. In differentembodiments, because of disjoint amplitude and phase equalization, therate of iteration to equalize the amplitude is slower than a rate ofchange of phases in the optical signal and slower that that the rate ofiteration for equalizing the phase of the signal, as described below.For example, for one iteration updating 305 the filter coefficients thereceiver performs several iterations for updating the phase.

Some embodiments of the invention are based on a realization that FIRcoefficients defining the amplitudes disturbance are vary over time.Also, there is a need to use multiple pilot signals to determine the FIRcoefficients accurately. Therefore, some embodiments use the trainingmode with transmitting continuous sequence pilot signals to increase theaccuracy of FIR filter 301 based on radiuses of the continuous sequenceof the pilot signals. In those embodiments, the training using thecontinuous sequence of pilot signals result in faster and more accurateinitialization of the FIR coefficients, which enables subsequenttracking of those FIR coefficients over time.

To that end, in some embodiments of the invention, the transmission ofthe optical signal includes a training mode and a decoding mode. Duringthe training mode, the digital signal includes a continuous sequence ofpilot symbols, the embodiments determine, during the training mode, thefiltering coefficients based on the error between the amplitudes of thereceived and the transmitted pilot symbols in the continuous sequenceand initialize the filtering coefficients determined during the trainingmode for use during the decoding mode.

For example, one embodiment determines an average error between theamplitudes of the received and the transmitted pilot symbols in thecontinuous sequence and determines the filtering coefficient using theaverage error. In different embodiments the average error is determinedusing a block and/or a sliding window update. During the block update,the average error is determined for the entire sequence or for eachsubset of the entire sequence of pilot symbols. During the slidingwindow the average error is determined for pilot symbols of the windowsliding from the start to the end of the sequence. Resulting averageerrors are used to determine the filtering coefficients. Additionally oralternatively, one embodiment determines an error between amplitude ofeach received pilot symbol and a corresponding transmitted pilot symbolsin the continuous sequence to produce a sequence of errors and updatesthe filtering coefficient iteratively for each error in the sequence oferrors.

In different embodiments, the average error is determined using the lowpass filter filtering out the noise in the error signal. This embodimentis based on the realization that the errors between the transmittedpilot symbols and the pilot symbols filtered by the filter withfiltering coefficients are caused by different sources. Those sourcesinclude channel noise and the mismatch between correct filtercoefficients and the coefficients currently used for the filteringoperation. Because the objective of initializing the filteringcoefficients does not consider compensating for the noise, the low passfilter is designed to cancel the noise effect in determining the averageerror. In some embodiments, the noise effect is reduced using the lowpass filter.

FIG. 4 shows a schematic of the receiver for a pilot-aided opticalcommunication system that uses the training and decoding modes accordingto one embodiment of the invention. An input signal (400) is preparedfor processing (401), by performing static functions such asnormalization, chromatic dispersion compensation, andintradyne-frequency offset compensation. The signal is optionally sentto a training mode (410), which aims to perform accurate initializationof the equalization, carrier phase estimation (CPE) and demodulationalgorithms. In training mode, the signal is sent to a trained DP-CMAadaptive equalizer (411). A training sequence (414) of known symbols isused to calculate the CMA error terms, from which the tap update of theDP-CMA algorithm is calculated (415), using for example, the LMS method.

The output of the equalizer has randomly varying phase, which is trackedby a trained CPE algorithm (412), which also takes advantage of theknown training sequence (414). The output of the trained CPE is used tocalculate the centroid of each constellation point in the trainingsequence (413). When the equalizer has converged sufficiently well, andthe centroid calculations are considered accurate, the receiver isswitched to pilot-aided mode (420). In this case, the prepared signal isprocessed with a pilot-aided DP-CMA (421), whose taps are initializedoptionally by those calculated by the trained DP-CMA (411). Thepilot-aided DP-CMA equalizer is adapted periodically (425) according tothe pilot sequence (424), although the update takes account of theradius of the pilot symbols only. After equalization, the signalundergoes pilot-aided CPE (422), with the initial estimate of phasebeing optionally provided by the training mode. Following CPE, thesignal undergoes demodulation and decoding (423), with these operationsoptionally accounting for the centroid calculations provided by (413)during training mode. Finally, the recovered and decoded data are sentto its destination, e.g., the sink (426).

Some of these embodiments are based on the realization that accuratetraining of the amplitude equalizer (421) is required due to thesensitivity of densely modulated signals to noise. Furthermore, thespeed of convergence for the equalizer can be improved if the sequenceis entirely known to the receiver, i.e., that is, a training sequence(414), rather than a sequence of symbols, of which only a partial subsetis known to the receiver, i.e., the pilot sequence (424). Otherembodiments of the invention are based on the realization that somedistortion on the signal is static and nonlinear (such as distortion dueto imperfect modulator biasing), and it is possible to improve theaccuracy of both carrier phase estimation (422), and demodulation (423)based on an analysis of the signal resulting from the training sequence(413). For example, the nonlinear distortion is trained by the LMS orRLS algorithms to analyze data-dependent statistics of the centroids,covariance, skewness, and correlation using the training sequence.

Phase Equalization

Some embodiments of the invention are based on a realization that due tothe physical nature of the optical channel, the phases of the receivedpilot symbols depend on phases of other received symbols including otherpilot symbols. Therefore, by collectively considering the multitude ofphases of the received pilot symbols, the dependency can be used toestimate the phase of the data symbols.

Other embodiments of the invention are based on the realization that dueto the nature of the optical channel, the transmitted and received pilotsymbols can be used to calculate not only an initial phase estimate, butalso a distribution of probabilities of time-varying phases.

Therefore, by exploiting not only the most likely initial phaseestimate, but also the initial probability distribution of the phases,the performance of phase equalization can be improved. In addition, theestimated probability information can provide more reliable calculationof soft-decision information at the demodulator so that the FEC decodercan efficiently correct potential errors after equalizations.

FIG. 5A shows a block diagram of a method for pilot-aided phaseequalization of the transmitted optical signal according to someembodiments of the invention. The method determines 501 a probabilitydistribution of phase noise on the pilot symbols using a statisticalprobability distribution 507 of phase noise on the optical channel anderrors 506 between phases of the received and the transmitted pilotsignals. Based on the determined probability distribution of phase noiseon the pilot symbols, the method determines 502 a probabilitydistribution of phase noise on the data symbols and demodulates 503 thefiltered signal using the probability distribution of phase noise on thedata symbols.

For example, one embodiment determines phase noise corresponding to theprobability distribution of phase noise on the data symbols and applies504 a phase shift equal to an opposite of the phase noise to thefiltered signal. Additionally or alternatively, one embodiment applies505 the probability distribution of phase noise on the data symbols tolog-likelihood ratio (LLR) calculations for the demodulation.

FIG. 5B shows a block diagram of an exemplar implementation ofpilot-aided phase equalization according to one embodiment of theinvention. In this example, four pilot symbols (515, 516, 517, and 518)are used to estimate phase on one block of data symbols (510). Thepilots, which can include adjacent and/or non-adjacent pilot symbols,are used to calculate the approximate posterior probability distributionof pilot phases (514). These phases are then used to perform initialestimation of the probability distributions of data symbol phases (511),given statistical models of phase noise and additive noise. These datasymbol phase estimates are then refined (512), and filtered (513),before the final phase estimates (519) are output.

FIG. 5C shows a block diagram of a method for determining theprobability distribution of phase noise on the pilot symbols accordingto one embodiment of the invention. The method determines 521 means ofthe probability distribution of phase noise on the pilot symbols usingthe errors between phases of the received and the transmitted pilotsignals and determines 522 variances of the probability distribution ofphase noise on the pilot symbols using variances of the statisticalprobability distribution of phase noise and distortion from the opticalchannel.

In some variations, the method also filters 523 the means and thevariances of the probability distribution of phase noise on the pilotsymbols to reduce a distortion of the means and the variances. Forexample, the method can use forward and/or backward Kalman filter forfiltering the posterior means and variances of the probabilitydistribution of phase noise. In another example, the method can useWiener filter to minimize the mean-square error given the phase noisemodel such as a Wiener process.

FIG. 5D shows a block diagram of a method for refining the probabilitydistribution of phase noise on the data symbols according to aprobability distribution of the data symbols and the received datasymbols according to one embodiment of the invention. Initial estimates530 of data symbol phases are the input to the method. Afterinitialization (531), symbol likelihoods are calculated in parallel foreach received data symbol (532). To calculate the symbol likelihoods,the initial phase estimates of means and variances are interpolated tocalculate symbol likelihoods for each received data symbol. For example,linear interpolation, second-order polynomial interpolation, cubicspline interpolation, or Gaussian process interpolation is used. Eachphase estimate is then updated according to the symbol likelihoods andreceived signal (533). If the maximum iteration count is reached, themethod then terminates (534), otherwise, the symbol likelihoods arere-calculated using the new phase estimates. The final output (535) isthe phase estimate after expectation-maximization (EM).

FIG. 5E shows a block diagram of a method for filtering the refinedprobability distribution of phase noise on the data symbols to produce afinal estimate of the probability distribution of phase noise on thedata symbols according to one embodiment of the invention. The inputsymbol phases (540), are filtered with some low-pass filter; such as arectangular, averaging filter (541). The filtered phase estimates arethen output (542).

Some embodiments of the invention are based on realization that a subsetof corresponding received and transmitted pilot signals can be groupedtogether to form a group, and that group of the pilot symbols can beused for determining the average error of the amplitudes for determiningthe filtering coefficients and/or for determining the errors between thephases of the received and the transmitted pilot symbols in the group todetermine the probability distribution of phase noise on the pilotsymbols.

Moreover, some embodiments are based on a recognition that such a groupcan be formed by pilot symbols received at different instance of time onthe optical channel, as shown in FIG. 1B, but also such a group can beformed by pilot symbols received on different optical channels.

FIG. 6 shows the structure of the pilot symbols within the data symbolsfor the purpose of multi-channel phase estimation. Several pilot symbols(611, 612, 613, 621, 622, 623, 641, 642, 643, 651, 652, 653) aredistributed through several blocks of data symbols (631, 632, 633) onseveral different channels. Phase estimation on a several, simultaneousblocks of data symbols (631, 632, 633) can be performed jointly, usingadjacent pilot symbols (621, 622, 623, 641, 642, 643) and optionally,non-adjacent pilot symbols (611, 612, 613, 651, 652, 653).

FIG. 7A shows a block diagram of a transmitter for a multi-channelpilot-aided optical communications system according to some embodimentsof the invention. Data (702) from a source (701), are sent to a datade-multiplexer (DM) (710), where the data are broken up into severaldata streams, each of which is to be transmitted on a differentwavelength. Each of the data streams is then sent to its own transmitter(720, 730, 740). In the transmitter, the data are encoded by an FECencoder (721, 731, 741), before pilot symbols (724, 734, 744) areinserted intermittently. The signals then undergo processing with DSPand other front end electronics (722, 732, 742) such asanalog-to-digital convertors. The signals are then sent to thetransmitter optics (723, 733, 743) for modulation on to the opticalcarrier. The optical signal is then sent to a wavelength multiplexer(WM) (750), to optionally being combined with other independent opticalsignals which have different wavelengths (751), before being sent to theoptical channel (755).

FIG. 7B shows a block diagram of a receiver for a multi-channelpilot-aided optical communications system according to some embodimentsof the invention. The signal from the optical channel (760) is sent to awavelength de-multiplexer (WDM) (761). Other wavelengths (762) areoptionally sent to other receivers for processing independently of thewavelength channel of interest. The signals of interest are then sent tothe relevant sub-channel receivers (770, 791). Inside the receiver, theoptical signal is detected by the optical receiver front end (771). Thisblock comprises both optical and electronic elements, such asdown-conversion, amplification, and quantization. The digital signal isthen processed by DSP (772).

After DSP processing, the received pilot symbols are extracted (774),and processed (775) in combination with the known transmitted pilotsequence (776). The extracted pilot symbols (774) are also sent to aprocessor which considers pilot symbols from all sub-channels (780). Inthis processor, received pilot symbols are aggregated from differentsub-channels (781), before digital processing is performed (782),considering both received and transmitted pilot sequences from allsub-channels. Information from both the single channel pilot processing(775) and multi-channel pilot processing (782) is then used in the RxDSP (772) for functions such as amplitude and phase equalization. Theprocessed signal is then sent for demodulation and FEC decoding (773),before being re-multiplexed (790), and sent to the data sink (795).

FIG. 8 shows a schematic of the receiver DSP for a multi-channelpilot-aided optical communications system incorporating training-aidedinitialization according to some embodiments of the invention. The inputsignals (830) are prepared for processing (831), by performing staticfunctions such as normalization, chromatic dispersion compensation, andintradyne-frequency offset compensation. The signals are optionally sentto a training mode (840), which aims to perform accurate initializationof the equalization, carrier phase estimation (CPE) and demodulationalgorithms. In training mode, the signal is sent to N-parallel, trainedDP-CMA adaptive equalizers (841). N-parallel training sequences (844) ofpilot symbols are used to calculate the CMA error terms, from which thetap updates of the DP-CMA algorithms are calculated (845), using forexample, the LMS algorithm. The outputs of the equalizers have randomlyvarying phase, which is tracked by trained CPE algorithms (842), whichalso take advantage of the known training sequences (844).

The outputs of the trained CPE are used to calculate the centroid ofeach constellation point in the training sequence (843). When theequalizer has converged sufficiently well, and the centroid calculationsare considered accurate, the receiver is switched to pilot-aided mode(850). In this case, the prepared signals are processed with N-parallelpilot-aided DP-CMA algorithm equalizers (851), whose taps areinitialized optionally by those calculated by the trained DP-CMAequalizers (841). The pilot-aided DP-CMA equalizers are adaptedperiodically (855) according to the pilot sequences (854), although theupdate takes account of the radius of the pilot symbols only.

After equalization, the signal undergoes pilot-aided joint CPE (852),where information from received pilot symbols on all sub-channels areconsidered in the CPE algorithm. The initial estimates of phase areoptionally provided by the training mode. Following CPE, the signalsundergo demodulation (853) and decoding (856), with these operationsoptionally accounting for the centroid calculations provided by (843)during training mode. Finally, the recovered and decoded data are sentto its destination, the sink (857).

Exemplar Embodiment Setup

N information symbols are transmitted in a block (510 in FIG. 5B). Toestimate the phase of a symbol transmitted during the n-th signalinginterval, K₁ pilots preceding and K₂ pilots following the consideredsymbol are used. Without loss of generality, assume that K₁=K₂=K.Therefore, the phases of information symbols belonging to the same blockare estimated using the same set of pilots

={p₁, . . . , p_(K), p_(K+1), . . . p_(2K)} such that the pilots p_(K)and p_(K+1) border the considered information block. FIG. 5B shows anexample where K=2 and phase estimation of data symbols is aided withpilots p₁, p₂, p₃ and p₄ (515, 516, 517 and 518 in FIG. 5B). In someembodiments of pilot-aided phase recovery schemes use K=1. Additionallyor alternatively, a single pilot might belong to more than one set ofpilots. Also, the phases corresponding to different information blockscan be estimated using different sets of pilots.

Assuming all signal impairments but phase and additive noise have beencompensated, a sample of the received signal at discrete time n, y_(n),is related to the symbol transmitted in the corresponding signalinginterval, x_(n), as

y _(n) =X _(n) e ^(jθ) ^(n) +ν_(n),  (1)

where θ_(n) and ν_(n) are the samples of respectively, a real-valuedphase noise and complex-valued additive white Gaussian noise. That is,ν_(n)˜

(0, σ²), while phase noise θ_(n) is modeled in 507 (FIGS. 5A and 5B) asa Wiener process

θ_(n)−θ_(n-1)˜

(0,σ_(p) ²), σ=_(p) ² =πΔνT _(s),  (2)

where Δν is an effective total linewidth of the transmitter's andreceiver's lasers and T_(s) is the signaling interval (inverse of thebaud rate).

Because the consecutive pilots p_(k+1) and p_(k) are separated by N+1signaling intervals (i.e., by N information symbols), the phase changeis using (2) modeled as

θ_(p) _(k+1) −θ_(p) _(k) <

(0,(N+1)σ_(p) ²), where k=1, . . . , 2K−1.  (3)

From Equation (1), the distribution of the received signal y_(n),conditioned on the transmitted symbol x_(n) and phase noise θ_(n), isgiven by

p(y _(n) |x _(n),θ_(n))˜

(x _(n) e ^(jθ) ^(n) ,σ²).  (4)

The phase recovery is framed as a statistical inference problem and thegoal is to compute/approximate the probability distribution of unknownphase θ_(n), conditioned on our knowledge of transmitted symbols andreceived signals at pilot locations. The method approximates

p(θ_(n) |x _(p) _(k) ,y _(p) _(k) ,k=1, . . . ,2K),n=1, . . . ,N.

Given the transmitted symbols and received signals at pilot locations,the method infers phases of pilot symbols in 501 (FIG. 5A) and 514 (FIG.5B). Using the inferred pilot symbol phases and the received signals atthe locations of the data symbols, the method estimates phases ofinformation symbols in 502 (FIG. 5A and FIG. 5B). The output includesthe estimates of information symbol phases and (optionally) soft andhard estimates of the transmitted symbols.

Determining Probability Distribution of Phase Noise on the Pilot Symbols

The goal of this processing stage is to infer pilot symbol phases basedon transmitted symbols and received signals on pilot locations.Formally, this stage evaluates the posterior distributionp(θ_(pk)|x_(p1), y_(p1), . . . , x_(p2K), y_(p2K)), k=1, . . . , 2K. Ablock diagram summarizing processing steps is shown in FIG. 5C.

Processing in this stage starts with computing the posteriordistribution p(θ_(pk)|x_(pk),y_(pk)). This posterior can be evaluatedusing the Bayes' rule and model (1). However, this distribution is notgiven in a closed form and one embodiment approximates the distribution.The Laplace method approximates a probability distribution with Gaussiandistribution, of mean equal to the maximum likelihood estimate of theunderlying parameter, and variance evaluated from the observed Fisherinformation.

As used by some embodiments after few steps of derivations, which areomitted here, the pilot symbol phases are approximated in 514 (FIG. 5B)as

p(θ_(p) _(k) |x _(p) _(k) ,y _(p) _(k) )˜

(μ_(p) _(k) ,σ_(p) _(k) ²), k=1, . . . ,2K,  (5)

where means and variances

$\begin{matrix}{{\mu_{p_{k}} = {\arg \left\{ {y_{p_{k}}x_{p_{k}}^{*}} \right\}}}{and}{\sigma_{p_{k}}^{2} = \frac{\sigma^{2}}{2{{x_{p_{k}}y_{p_{k}}}}}}} & (6)\end{matrix}$

are computed in 521 and 522 (FIG. 5C), respectively. For example, themean and variance of the approximating Gaussian distribution can beevaluated for each pilot separately and in parallel. Such an evaluationresults in a closed form expressions for mean and variance. In addition,Gaussian distribution with this mean and variance accuratelyapproximates the true posterior distribution.

The posterior distributions evaluated in the previous step are updatedso as to account for correlations between phases of pilot symbols. Thisis achieved by using Kalman filtering framework in 523 (FIG. 5C). Thelinear dynamical model is the Wiener model for phase noise dynamics (2).Using the result from the previous step, the observation model isconstructed as

ψ_(p) _(k) =θ_(p) _(k) +n_(p) _(k) , where ψ_(p) _(k) =μ_(p) _(k) andn_(p) _(k) ˜

(0,σ_(p) _(k) ²).  (7)

Note that the mean μ_(p) _(k) is the “observation” of an unknown phaseψ_(p) _(k) . The observation noise n_(p) _(k) is Gaussian distributedwith zero mean and variance σ_(p) _(k) ². The means and variances areevaluated in the previous step using (6).

Using linear dynamical model (2) and observation model (7), Kalmanfilter (i.e., full forward pass through the model) yields

p(θ_(p) _(k) |x _(p) ₁ ,y _(p) ₁ , . . . ,x _(p) _(k) ,y _(p) _(k) )˜

({tilde over (μ)}_(p) _(k) ,{tilde over (σ)}_(p) _(k) ²),k=1, . . .,2K.  (8)

This is followed by Kalman smoother which performs backward pass frompilot p_(2K) up to and including pilot p_(K+1) and yields means ofGaussian posteriors. That is,

p(θ_(p) _(k) |x _(p) ₁ ,y _(p) ₁ , . . . ,x _(p) _(k) ,y _(p) _(k) )˜

(ν_(p) _(k) ,.),k=K+1, . . . ,2K.  (9)

The processing step in 523 involving Kalman filtering and smoothing canrequire sequential processing. To overcome this shortcoming, the numberof pilots 2K can be reduced. In fact, some embodiments are based onunderstanding that increasing the number of pilots 2K beyond some smallnumber provides no performance gain. This threshold depends on thenumber of information symbols in a block N and variance of phase noisejumps σ_(p) ² and can very likely be as low as 4.

Alternatively, the equivalent processing in this step can also beperformed in parallel by Wiener filtering in some embodiments withcomputational resources allowing for inverting a matrix of size 2K ateach pilot location. This can be advantageous because sequential Kalmansmoothing is not required and a small number of pilots of 2K=4 alreadybrings the solution to the edge of possible performance improvements inmany practical applications. In one embodiment, the matrix inversion isavoided by providing the inverse matrix of the auto-covariance matrixoffline based on the model of Wiener process.

The described processing stage outputs posterior means {tilde over(μ)}_(p) _(k) and ν_(p) _(K+1) and posterior variance {tilde over(σ)}_(p) _(K) ². These quantities contain information required for phaseestimation of data symbols in 502, described in the following part.

The pseudo-code corresponding to the described processing stage is:Data: Transmitted and received pilots: x_(p) ₁ , y_(p) ₁ , . . . , x_(p)_(2K) , y_(p) _(2K)Result: Posterior means and variance: {tilde over (μ)}_(p) _(K) , ν_(p)_(K+1) and {tilde over (σ)}_(p) _(K) ²Parallel for k=1: 2K do

${\mu_{p_{k}} = {{\arg \left\{ {y_{p_{k}}x_{p_{k}}^{*}} \right\} \mspace{14mu} {and}\mspace{14mu} \sigma_{p_{k}}^{2}} = \frac{\sigma^{2}}{2{{x_{p_{k}}y_{p_{k}}}}}}};$end${{\overset{\sim}{\mu}}_{p_{1}} = {{\mu_{p_{1}}\mspace{14mu} {and}\mspace{14mu} {\overset{\sim}{\sigma}}_{p_{1}}^{2}} = \sigma_{p_{1}}^{2}}};$for  k = 1:  2 K  do${b = {{{\left( {N + 1} \right)\sigma_{p}^{2}} + {{\overset{\sim}{\sigma}}_{p_{k - 1}}^{2}\mspace{14mu} {and}\mspace{14mu} a}} = \frac{b}{b + \sigma_{p_{k}}^{2}}}};$${{\overset{\sim}{\mu}}_{p_{k}} = {{{a\; \mu_{p_{k}}} + {\left( {1 - a} \right){\overset{\sim}{\mu}}_{p_{k - 1}}\mspace{14mu} {and}\mspace{14mu} {\overset{\sim}{\sigma}}_{p_{k}}^{2}}} = {a\; \sigma_{p_{k}}^{2}}}};$end ${v_{p_{2\; K}} = {\overset{\sim}{\mu}}_{p_{2\; K}}};$for  k = 2 K:  (−1):  (K + 1)  do${a = \frac{{\overset{\sim}{\sigma}}_{p_{k}}^{2}}{{\overset{\sim}{\sigma}}_{p_{k}}^{2} + {\left( {N + 1} \right)\sigma_{p}^{2}}}};$${v_{p_{k}} = {{a\; v_{p_{k + 1}}} + {\left( {1 - a} \right){\overset{\sim}{\mu}}_{p_{k}}}}};$end.

Determining Probability Distribution of Phase Noise on the Data Symbols

This stage, which performs phase estimation of data symbols, can includethree processing steps 511, 512 and 513, shown in FIG. 5B according tosome embodiments. The processing step 511 delivers initial estimates ofdata symbol phases using the posterior distributions of pilot symbolphases, evaluated in 501 (FIG. 5A). The information symbols are locatedbetween pilots p_(K) and p_(K+1), and using the Wiener process model forphase noise (2), it can be shown that the posterior p(θ_(n)|x_(p) ₁ ,y_(p) ₁ , . . . , x_(p) _(2K) , y_(p) _(2K) ), n=1, . . . , N, isGaussian distributed with mean and variance dependent upon means andvariances of Gaussian posteriors corresponding to pilots p_(K) andp_(K+1). More precisely, omitting the derivation details, posteriordistribution of the n-th information symbol is given by

p(θ_(n) |x _(p) ₁ ,y _(p) ₁ , . . . ,x _(p) _(2K) ,y _(p) _(2K) )˜

(μ_(n),σ_(n) ²),  (10)

where the mean μ_(n) is given by a linear interpolation:

$\begin{matrix}{\mu_{n} = \frac{{\left( {N + 1 - n} \right)\sigma_{p}^{2}{\overset{\sim}{\mu}}_{p_{K}}} + {\left( {{n\; \sigma_{p}^{2}} + {\overset{\sim}{\sigma}}_{p_{K}}^{2}} \right)v_{p_{K + 1}}}}{{\left( {N + 1} \right)\sigma_{p}^{2}} + {\overset{\sim}{\sigma}}_{p_{K}}^{2}}} & (11)\end{matrix}$

The posterior means μ_(n)'s are evaluated separately in parallel using(11). They are viewed as initial estimates of the information symbolphases and are refined in the following processing step. In anotherembodiment, the mean and variances are obtained by a second-orderpolynomial interpolation, cubic spline interpolation, and Gaussianprocess Kriging interpolation in a parallel fashion.

The initial phase estimates are refined in 512 by employing the EMmethod, as outlined in FIG. 5D. Due to a non-convex nature of theunderlying optimization problem, the EM method converges to a localstationary point closest to the initial point. Therefore, the EM methodneeds to be initialized with a phase estimate that is already reasonablyclose to the true phase to yield better phase estimate. Otherwise, themethod converges to some other, undesirable stationary point.

A separate EM method refines initial phase estimate μ_(n) of eachinformation symbol in parallel. In the following, we present thecomputations involved and skip the derivation details. The EM routinecorresponding to symbol x_(n) is initialized with {circumflex over(θ)}_(n) ⁽⁰⁾=μ_(n). The k-th iteration starts with evaluating thelikelihood of symbol x_(n) given the received signal y_(n) and phaseestimate θ_(n) ^((k−1)), obtained from iteration k−1. This likelihood,evaluated in 532 (FIG. 5D), is up to a normalization constant given by

$\begin{matrix}{{p\left( {{x_{n} = {ay_{n}}};{\hat{\theta}}_{n}^{({k - 1})}} \right)} \propto {{p\left( {{{y_{n}x_{n}} = a};{\hat{\theta}}_{n}^{({k - 1})}} \right)}{p\left( {{x_{n} = a};{\hat{\theta}}_{n}^{({k - 1})}} \right)}}} & (12) \\{{\propto {\exp \left( {{- \frac{1}{\sigma^{2}}}{{y_{n} - {a\; ^{j\; {\hat{\theta}}_{n}^{({k - 1})}}}}}^{2}} \right)}},} & (13)\end{matrix}$

where x_(n) takes values from the transmitted constellation, i.e., aεχ.The transmitted symbols are uniformly at random drawn from theconstellation so that p(x_(n)=a; {circumflex over (θ)}_(n) ^((k−1)))∂1.The symbol likelihoods are further used to update phase estimate in 533(FIG. 5D) such that

{circumflex over (θ)}_(n) ^(k)=arg(y _(n)Σ_(αεχ) a*p(x _(n) =a|y_(n);{circumflex over (θ)}_(n) ^((k−1))))  (14)

The EM method is performed until a termination condition 534 (FIG. 5D)is satisfied, e.g., until a predefined number of iterations I_(max) isreached and outputs EM phase estimates in 535 (FIG. 5D). To reduce thecomputational complexity, the number of iterations I_(max) can be keptsmall. In some embodiments, the method converges after only twoiterations and no improvement is made if more than two iterations areused.

Additionally, the complexity burden arising from computing the symbollikelihoods in high order modulation formats (such as 64 QAM or 256 QAM)can be alleviated by considering only a subset of symbols in theconstellation. For example, the likelihoods of symbol x_(n) can beevaluated only for a certain number of symbols in the constellationwhich are closest in the Euclidean distance to y_(n)e^(−jμ) ^(n) . Thisis because μ_(n) is already relatively close to the true phase θ_(n)which, in the first line, makes the application of the EM procedure aviable approach.

The EM methods can be performed separately on information symbols (andthus in parallel). This, in turn, means that the correlation structureof phase variations across symbols is not exploited. Some embodimentsdevelop the EM method which takes into account the statistics of phasevariations. However, the phase estimates in such a method are updated asarguments which minimize some objective function and are not given inclosed forms. Also, such a method does not admit parallel implementationand is therefore not practical.

To overcome the shortcoming of not taking into account the statistics ofphase variations in the EM procedures, the final processing step 513filters the EM phase estimates {circumflex over (θ)}_(n) ^(I) ^(max) ,as outlines in FIG. 5E. For example, one embodiments filter theseestimates using a moving average filter such that the final phaseestimate at discrete time n is

$\begin{matrix}{{\hat{\theta}}_{n} = {\frac{1}{{2\; T} + 1}{\sum\limits_{i = {n - T}}^{n + T}\; {{\hat{\theta}}_{i}^{(I_{\max})}.}}}} & (15)\end{matrix}$

The method outputs phase estimates 519 of the data block.

Optionally, along with the final phase estimates, the method outputssoft and hard decisions of the transmitted symbols. Namely, given thefinal phase estimate {circumflex over (θ)}_(n) and received signaly_(n), the soft decision is given as a vector of likelihoods evaluatedfor all constellation points aεχ. It is up to a normalization constantgiven by

$\begin{matrix}{{p\left( {{x_{n} = {ay_{n}}};{\hat{\theta}}_{n}} \right)} \propto {{\exp \left( {{- \frac{1}{\sigma^{2}}}{{{y_{n}^{{- j}\; {\hat{\theta}}_{n}}} - a}}^{2}} \right)}.}} & (16)\end{matrix}$

and evaluated in 505 (FIG. 5A). The hard decision is a constellationpoint a which maximizes the symbol likelihood evaluated above. Inanother embodiment, the residual phase noise estimation error based onthe phase variance is used to calculate more accurate symbollikelihoods.

For example, the likelihood in (16) can be modified by the Tikhonovdistribution, which uses the zero-th order Bessel function of the firstkind to take the residual phase noise into account. In another example,the likelihood is modified by a linear-transform and bilinear transformto take the residual phase noise variance into account. This modifiedlikelihood is advantageous when the pilot symbol interval is large andlow SNR regimes because the residual phase noise variance is notconstant over the data symbols.

The pseudo-code corresponding to the described processing stage is:

Data: Received signals: y₁, . . . , y_(N)Input: {tilde over (μ)}_(p) _(K) , ν_(p) _(K+1) and {tilde over (σ)}_(p)_(K) ²Result: Final phase estimates: {circumflex over (θ)}₁, . . . ,{circumflex over (θ)}_(N)Parallel for n=1: N doα=(N+1−n)σ_(p) ², β=nσ_(p) ²+{tilde over (σ)}_(p) _(K) ², γ=α/(α+β);μ_(n)=γ{tilde over (μ)}_(p) _(K) +(1−γ)ν_(p) _(K+1) ;{tilde over (θ)}_(n) ⁽⁰⁾=μ_(n);for k=1:I_(max) dop(x_(n)=a|y_(n);{circumflex over (θ)}_(n) ^((k−1)))∂exp(−σ⁻²y_(n)−ae^(j{circumflex over (θ)}) ^(n) ^((k−1))|²), aεχ;{circumflex over (θ)}_(n)^((k))=arg(y_(n)Σ_(aεχ)a*p(x_(n)=a|y_(n);{circumflex over (θ)}_(n)^((k−1))));endendFilter {circumflex over (θ)}₁ ^(I) ^(max) , . . . , {circumflex over(θ)}_(N) ^((I) ^(max) ⁾ to N yield {circumflex over (θ)}₁, . . . ,{circumflex over (θ)}_(N).

Multi-Channel Phase Estimation

Some embodiments perform phase estimation in a multi-channelcommunications system. Multiple channels can refer to different carriersin a multi-carrier transmission, or different polarizations in a singlecarrier transmission, or different polarizations of multiple carriers.Multi-channel can be optionally extended to multi-mode multiplexingsystems in a straightforward manner.

The overall number of channels, irrespective of the underlyingmechanism, which generates the channels, is denoted with L. The signalreceived in channel l and at discrete time n is after equalization givenby

y _(n) ^((l)) =x _(n) ^((l)) e ^(jθ) ^(n) ^((l)) +ν_(n) ^((l)),  (17)

where l=1, . . . , L, x_(n) ^((l)) is the transmitted symbol, θ_(n)^((l)) is a sample of phase noise, and ν_(n) ^((l)) is a sample ofadditive noise, all corresponding to channel l and discrete time n. Theadditive noise is complex-valued, zero mean, white Gaussian process ofvariance dependent upon channel index, i.e., ν_(n) ^((l))˜

(0, σ_(l) ²).

As in a single channel case, each channel in a multi-channel schemetransmits a block of N information symbols, preceded by a pilot symbol,as shown in FIG. 6. The phase estimation of information symbols within ablock is aided with K pilots preceding and K pilots following the block.In general, the number of pilots used on each side and in each channelcan be different.

Some embodiments analyze different channels separately and apply themethod described in the previous part to each channel in parallel.However, if phase variations across channels are correlated, theestimation performance can benefit from joint phase estimation. In thefollowing, three possible embodiments for corresponding joint phaseestimation are described.

Constant Phase Offsets Between Channels

In the first embodiment, phases of any two channels are equal up to someconstant phase offset. Formally, the carrier phase of channel l isrepresented as

θ_(n) ^((l))=θ_(n)+Δθ^((l)),  (18)

where θ_(n) is the phase of the reference channel and Δθ^((l)) is thephase offset with respect to the reference channel. The phase offsetdepends on the channel index l and is constant in time n. Assuming thatthe phase offsets Δθ^((l)) are given or estimated in some way, the goalis to estimate the common phase θ_(n), n=1, . . . , N.

To bring all the channels to the same phase, the phase offsets arecompensated by rotating the received signals such that

{tilde over (y)} _(n) ^((l)) =y _(n) ^((l)) e ^(−jΔθ) ^((l)) ,l=1, . . .,L.  (19)

This processing can be done in parallel and the resulting symbolsequences share the same phase across channels. This phase is estimatedby employing a similar procedure as used for a single channel case. Theonly difference is that now a group of L pilots (one for each channel)which effectively undergo the same phase shift are used to estimate thephase at the corresponding time instant.

The method starts with approximating the posterior of pilot symbol phaseθ_(p) _(k) with Gaussian distribution whose mean and variance areevaluated in respectively 521 and 522 (FIG. 5C) by taking advantage ofall pilot symbols across L channels that correspond to the same instantp_(k) and are given by

$\begin{matrix}{{\mu_{p_{k}} = {\arg \left( {\sum\limits_{l = 1}^{L}\; {{\overset{\sim}{y}}_{p_{k}}^{(l)}\left( x_{p_{k}}^{(l)} \right)}^{*}} \right)}}{and}{\sigma_{p_{k}}^{2} = {\frac{0.5}{\sum\limits_{l = 1}^{L}\; \frac{{x_{p_{k}}^{(l)}{\overset{\sim}{y}}_{p_{k}}^{(l)}}}{\sigma_{l}^{2}}}.}}} & (20)\end{matrix}$

The pilot phase posteriors are processed through a cascade of Kalmanfilter (with full forward pass) and Kalman smoother (with backward passfrom p_(2K) up to and including p_(K+1)) in 523 (FIG. 5C) the same wayas for a single channel case. The smoothed pilot phase estimates areused to obtain initial estimates of information symbol phases using thesame expression as for a single channel case.

The initial phase estimates of information symbols are refined in 512(FIG. 5B) using the EM procedure, as outlined in FIG. 5D. The k-th EMiteration starts with evaluating the likelihoods of symbols across Lchannels in 532 (FIG. 5D). These likelihoods are evaluated for eachsymbol separately in parallel using

$\begin{matrix}{{{p\left( {{x_{n}^{(l)} = {a{\overset{\sim}{y}}_{n}^{(l)}}};\theta_{n}^{({k - 1})}} \right)} \propto {\exp \left( {{- \frac{1}{\sigma_{l}^{2}}}{{{\overset{\sim}{y}}_{n}^{(l)} - {a\; ^{j\; {\hat{\theta}}_{n}^{({k - 1})}}}}}^{2}} \right)}},{a \in X},} & (21)\end{matrix}$

where l=1, . . . , L and n=1, . . . , N. Note that the superscript withl refers to channel index, while the superscript with k refer toiteration index.

The k-th EM iteration delivers an updated phase estimate evaluated in533 (FIG. 5D) using

{circumflex over (θ)}_(n) ^((k))=arg(Σ_(l=1) ^(L) {tilde over (y)} _(n)^((l))Σ_(aεχ) a*p(x _(n) ^((l)) =a|y _(n) ^((l));{circumflex over(θ)}_(n) ^((k−1)))).  (22)

As for a single channel embodiment, the EM method does not exploitcorrelation structure of the phase process. Therefore, to overcome thisshortcoming, the EM phase estimates are filtered (for example, using amoving average filter) in 513 (FIG. 5B). The filter outputs are thefinal phase estimates 519 (FIG. 5B). Optionally, soft and hard decisionsof the transmitted symbols can be evaluated from the final phaseestimates and received signals.

Correlated Phases Across Channels

Another embodiment uses a method for joint phase estimation when phasechanges across channels are correlated. To formalize the model, denotewith θ_(i) a vector of phases at discrete time i across channels l=1, .. . , L. That is,

θ_(i)=[θ_(i) ^((l)) . . . θ_(i) ^((L))].^(T)  (23)

The phase vector θ_(i) varies such that the change between twoconsecutive time instants i−1 and i is modeled as

θ_(i)−θ_(i-1)˜

(0,C),  (24)

where C is the covariance matrix of phase jumps across channels. Byassumption, this matrix is predefined or estimated using some methods.Equation (24) implies that phase in a single channel l follows a Wienerprocess with the variance of phase jumps equal to the correspondingdiagonal element in C, denoted with c_(ll).

The joint estimation of phases across channels can be performed in twostages 501 and 502. The first stage starts with approximating theposteriors of pilot symbol phases with Gaussian distributions. The mean521 and variance 522 of the approximating Gaussian corresponding topilot p_(k) in channel l are given by

$\begin{matrix}{{\mu_{p_{k}}^{(l)} = {\arg \; \left( {y_{p_{k}}^{(l)}\left( x_{p_{k}}^{(l)} \right)}^{*} \right)}}{{{{and}\left( \sigma_{p_{k}}^{2} \right)}^{(l)} = \frac{\sigma_{l}^{2}}{2{{x_{p_{k}}^{(l)}y_{p_{k}}^{(l)}}}}},}} & (25)\end{matrix}$

where x_(p) _(k) ^((l)) and y_(p) _(k) ^((l)) are, respectively, thetransmitted symbol and received signal corresponding to pilot p_(k) andchannel l, and σ_(l) ² is the variance of additive noise in channel l.The means μ_(p) _(k) ^((l)) across channels are collected into a columnvector μ_(p) _(k) =[μ_(p) _(k) ⁽¹⁾ . . . μ_(p) _(k) ^((L))]^(T).Similarly, the variances σ_(p) _(k) ^((l)) across channels form adiagonal matrix Σ_(p) _(k) =diag((σ_(p) _(k) ²)⁽¹⁾, . . . , (σ_(p) _(k)²)^((L))).

The initial estimates of pilot symbol phases are smoothed in 523 bytaking into account the correlation structure of phase variations intime and across channels. This is achieved by employing the Kalmanfiltering framework. Given that two consecutive pilots are separated byN information symbols, the linear dynamical model is using (24) given by

θ_(p) _(k+1) −θ_(p) _(k) ˜

(0,(N+1)C),k=1, . . . ,2K−1.  (26)

The observation model is constructed from the initial estimates of pilotsymbol phases obtained in (25) such that

ψ_(p) _(k) =θ_(p) _(k) +n _(p) _(k) ,k=1, . . . ,2K,  (27)

where the observed vector ψ_(p) _(k) =μ_(p) _(k) and observation noisen_(p) _(k) ˜

(0,Σ_(p) _(k) ).

Given the linear dynamical and observation model, the initial pilotphase estimates are processed via full forward pass of Kalman filteringand partial backward pass of Kalman smoothing that ends at pilotp_(K+1). The outputs of this processing stage are the mean vector {tildeover (μ)}_(p) _(K) and covariance matrix {tilde over (Σ)}_(p) _(K)corresponding to the pilot p_(K) (and obtained from the forward pass, aswell as the mean vector ν_(p) _(K+1) , corresponding to the pilotp_(K+1) and resulting from the backward pass.

Each step of sequential processing in Kalman filtering step performsmatrix inversion. The size of a matrix is equal to the number ofchannels L. To alleviate the computational complexity, one may reducethe number of pilots 2K aiding phase estimation. This number depends onthe number of information symbols N between two pilots and how quicklyphase varies in time. In some applications, using more than 2K=4 pilotsper channel provides no further gain in many scenarios of practicalinterest.

On the other hand, the complexity burden arising from performing matrixinversions might be reduced by, for example, approximating thecovariance matrix C with a tridiagonal matrix (meaning that only thephases in adjacent channels are correlated), in which case the matrixinverse is relatively easy to compute.

The outputs {tilde over (μ)}_(p) _(K) , ν_(p) _(K+1) and {tilde over(Σ)}_(p) _(K) from the first processing stage are used to obtain initialestimates of information symbol phases in 511. Conceptually, one caninterpolate between two Gaussian vectors (posteriors of the phases ofpilots p_(K) (and p_(K+1) across all channels). However, this wouldnecessitate computing N matrix inversions (one for each informationsymbol in a block). To alleviate this shortcoming, separate linearinterpolations between pilot phases p_(k) and p_(K+1) in each channelare performed, such that the initial phase estimate of symbol n inchannel l is given by

$\begin{matrix}{{\mu_{n}^{(l)} = \frac{{\left( {N + 1 - n} \right)c_{ll}{\overset{\sim}{\mu}}_{p_{K}}^{(l)}} + {\left( {{nc}_{ll} + \left( {\overset{\sim}{\sigma}}_{p_{K}}^{2} \right)^{(l)}} \right)v_{p_{K + 1}}^{(l)}}}{{\left( {N + 1} \right)c_{ll}} + \left( {\overset{\sim}{\sigma}}_{p_{K}}^{2} \right)^{(l)}}},} & (28)\end{matrix}$

Wherein c_(ll) is the l-th diagonal element of C which represents thevariance of phase noise jumps between two consecutive discrete timeinstants in the channel l. In another embodiment, second-orderpolynomial interpolation, cubic spline interpolation, or Gaussianprocess Kriging interpolation is used.

The initial phase estimates of information symbols are refined using theEM procedure 512. The EM procedure is separately applied to eachinformation symbol in each channel in parallel. The details are the sameas for a single channel case.

Note that phase correlations across channels are not taken into accountby performing separate EM procedures. Conceptually, the EM procedurescan be devised so as to account for these correlations. However, thiswould require more complicated routines for updating phase estimates.More specifically, a vector of phase estimates across channels would beupdated as an argument which minimizes some objective function and isnot given in a closed form.

The separate EM methods do not take into account phase correlations intime. In addition, even if computational resources allowed forperforming phase updates, such a scheme is not fully parallelizable. Thefinal phase estimates can be obtained by filtering 513 the EM phaseestimates in each channel separately. A moving average filter ispossible. Optionally, the soft and hard decisions of the transmittedsymbols are obtained from the final phase estimates and receivedsignals.

Highly Correlated Phases Across Channels

When phase variations across channels are highly correlated, the phaseoffsets between different channels slowly vary in time. This partpresents a joint phase estimation method which exploits thisobservation. More formally, the assumption is that phase offsets betweendifferent channels are constant during one information block.

The joint phase estimation method infers pilot phases in the same way asdescribed in the previous part. That is, it starts off withapproximating the posterior distributions of pilot phases and processesthem through a cascade of Kalman filtering and Kalman smoothing. Theonly difference is that full backward pass of Kalman smoothing isperformed. This yields means ν_(p) ₁ , . . . , ν_(p) _(2K) of Gaussianposteriors of 2K pilot phase vectors (recall that a phase vectorcollects phases across all L channels).

The posterior means obtained from the backward pass are used to estimatethe phase offsets between different channels. The phase offset betweenchannel l and some reference channel (without loss of generality, wechoose channel 1 to be a reference channel) is estimated by taking theaverage of the differences between corresponding pilot phase estimates,

$\begin{matrix}{{{\Delta \; \theta^{(l)}} = {\frac{1}{2\; K}{\sum\limits_{k = 1}^{2\; K}\; \left( {{v_{p_{k}}(l)} - {v_{p_{k}}(1)}} \right)}}},} & (29)\end{matrix}$

where ν_(pk) (l) is the l-th element of ν_(p) _(k) . Note that the aboveestimator uses all 2K pilots in both considered channels. However, ifphase offset varies on a shorted time scale, only a subset of pilotsshould be used. In the most extreme case, only pilots p_(K) and p_(K+1)are used.

After the phase offsets Δθ^((l))'s are determined, one embodimentrotates the information symbols in all channels using (19) so that theresulting symbols exhibit the same phase variation. The initial phaseestimates of information symbols are obtained from posterior means ofpilots p_(K) and p_(K+1) in the reference channel. The initial phaseestimates can be refined using the EM method using Equations (21) and(22). The resulting phase estimates are filtered, for example withmoving average filer, which yields the final phase estimates.Optionally, the soft and hard decisions of transmitted symbols areproduced based on final phase estimates and received signals.

The above three embodiments for multi-channel pilot-aided phaseestimation methods use all received signals obtained at differentchannels. In some embodiments, the received signals are shared amongdifferent channel receivers by using band-limited interconnects todistribute quantized received signals each other. To reduce the requireddata rates for interconnects, the received signals on pilot symbols areonly shared, or the phase estimates of pilot symbol posteriors obtainedby the single-channel pilot-aided algorithm are distributed to furtherrefine the phase estimates by taking the channel correlation intoaccount in multi-channel pilot-aided phase estimation algorithms. Insome embodiments, to decrease the required data rates for interconnects,cooperative quantization methods such as Wyner-Ziv coding are used. TheWyner-Ziv coding can reduce the amount of quantization data whilekeeping the quantization distortion low, by using the signal correlationover different channels.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

We claim:
 1. A method for decoding an optical signal transmitted over anoptical channel from a transmitter to a receiver, comprising: receivingthe transmitted optical signal; producing, from the transmitted opticalsignal, a digital signal including received data symbols and receivedpilot symbols; determining filtering coefficients based on an errorbetween amplitudes of the received pilot symbols and amplitudes oftransmitted pilot symbols, while ignoring errors between phases of thereceived pilot symbols and phases of the transmitted pilot symbols,wherein the amplitudes and the phases of the transmitted pilot symbolsare known at the transmitter and the receiver; filtering the digitalsignal according to the filtering coefficients to produce a filteredsignal with an equalized amplitude and an unconstrained phase; anddemodulating and decoding the filtered signal to produce an estimate ofthe transmitted optical signal, wherein at least some steps of themethod are performed using a processor of the receiver.
 2. The method ofclaim 1, wherein a transmission of the optical signal includes atraining mode and a decoding mode, wherein, during the training mode,the digital signal includes a continuous sequence of pilot symbols,further comprising: determining, during the training mode, the filteringcoefficients based on the error between the amplitudes of the receivedpilot symbols and the transmitted pilot symbols in the continuoussequence; and initializing, during the decoding mode, the filteringcoefficients determined during the training mode.
 3. The method of claim2, further comprising: determining an average error between theamplitudes of the received pilot symbols and the transmitted pilotsymbols in the continuous sequence using a low pass filter; anddetermining the filtering coefficient using the average error.
 4. Themethod of claim 2, further comprising: determining an error between eachamplitude of the received pilot symbol and a corresponding transmittedpilot symbols in the continuous sequence to produce a sequence oferrors; and updating the filtering coefficient iteratively for eacherror in the sequence of errors.
 5. The method of claim 1, furthercomprising: updating the filtering coefficients iteratively in responseto receiving each pilot symbol using a least-mean-square (LMS) orrecursive least-squares (RLS) update.
 6. The method of claim 1, furthercomprising: grouping a subset of corresponding received and transmittedpilot signals to form a group; determining an average error between theamplitudes of the received and the transmitted pilot symbols in thegroup; and determining the filtering coefficient using the averageerror.
 7. The method of claim 6, wherein the group is formed by pilotsymbols received at different instance of time on the optical channel.8. The method of claim 6, wherein the group is formed by pilot symbolsreceived on different optical channels.
 9. The method of claimed 1,further comprising: determining a probability distribution of phasenoise on the pilot symbols using a statistical probability distributionof phase noise on the optical channel and errors between phases of thereceived pilot symbols and the transmitted pilot signals; determining aprobability distribution of phase noise on the data symbols using thestatistical probability distribution of phase noise on the opticalchannel and the probability distribution of phase noise on the pilotsymbols; and demodulating the filtered signal using the probabilitydistribution of the phase noise on the data symbols.
 10. The method ofclaim 9, wherein the demodulating comprises: determining the phase noisecorresponding to the probability distribution of the phase noise on thedata symbols; and applying a phase shift equal to an opposite of thephase noise to the filtered signal.
 11. The method of claim 9, whereinthe demodulating comprises: applying the probability distribution ofphase noise on the data symbols to log-likelihood ratio calculations forthe demodulating.
 12. The method of claim 9, further comprising:refining the probability distribution of the phase noise in the datasymbols according to a probability distribution of the data symbols andthe received data symbols.
 13. The method of claim 12, furthercomprising: filtering the refined probability distribution of phasenoise on the data symbols to produce a final estimate of the probabilitydistribution of phase noise on the data symbols.
 14. The method of claim9, wherein the determining the probability distribution of the phasenoise on the pilot symbols comprises: determining means of theprobability distribution of the phase noise on the pilot symbols usingthe errors between phases of the received and the transmitted pilotsignals; and determining variances of the probability distribution ofthe phase noise on the pilot symbols using variances of the statisticalprobability distribution of phase noise and distortion from the opticalchannel.
 15. The method of claim 14, further comprising: filtering themeans and the variances of the probability distribution of phase noiseon the pilot symbols to reduce a distortion of the means and thevariances.
 16. The method of claim 9, further comprising: grouping asubset of corresponding received and transmitted pilot signals to form agroup; determining the errors between the phases of the received and thetransmitted pilot symbols in the group; and determining the probabilitydistribution of phase noise on the pilot symbols using the errorsdetermined for the group.
 17. The method of claim 16, wherein the groupis formed by pilot symbols received at different instance of time on theoptical channel.
 18. The method of claim 16, wherein the group is formedby pilot symbols received on different optical channels.
 19. A receiverfor decoding an optical signal transmitted by a transmitter over anoptical channel, comprising: a front end for receiving the transmittedoptical signal to produce a digital signal including data symbols andpilot symbols; an amplitude equalizer for determining filteringcoefficients based on an error between amplitudes of the received pilotsymbols and amplitudes of transmitted pilot symbols, while ignoringerrors between phases of the received pilot symbols and phases of thetransmitted pilot symbols and for filtering the digital signal accordingto the filtering coefficients to produce a filtered signal with anequalized amplitude and an unconstrained phase; a phase equalizer fordetermining a probability distribution of phase noise on the datasymbols using a statistical probability distribution of phase noise onthe optical channel and a probability distribution of phase noise on thepilot symbols; and a decoder for demodulating and decoding the filteredsignal using the probability distribution of phase noise on the datasymbols to produce an estimate of the transmitted optical signal. 20.The receiver of claim 19, wherein the amplitude equalizer and the phaseequalizer use a group of corresponding received and transmitted pilotsignals, wherein the group is formed by pilot symbols received atdifferent instance of time on the optical channel and by pilot symbolsreceived on different optical channels, wherein the received pilotsymbols are shared through band-limited interconnects with cooperativequantization.